Factoring Calculator

Factors of a number are the list of all integral numbers that can divide it evenly without leaving any remainders. For example, consider the number 16. The factors of 16 are:

1, 2, 4, 8, 16

16 is completely divisible by these numbers.

• 16 ÷ 1 = 10
• 16 ÷ 2 = 8
• 16 ÷ 4 = 4
• 16 ÷ 8 = 2
• 16 ÷ 16 = 1

Example Factors

All integers have at least two factors, the number 1 and itself. A number that has only two factors is only divisible by itself and 1. These numbers are called prime numbers. Conversely, all numbers with more than two factors are composite numbers.

Factor Pairs

Factor pairs are the combination of two factors that give the original number when multiplied together. For example, 16 has the following factor pairs:

How to perform factorization

You find out factors of a number by using a trial division. Let's say our number is n.

1. Step 1: Find the square root of n and round it down to the nearest whole number. Let's call this number r. This square root helps us reduce the calculation.
2. Step 2: Begin the trial division with the number 1. All numbers are completely divisible by 1. So, 1 is one of the factors by default. Add the pair (1, n) to our factor pair list.
3. Step 3: Repeat the above process for 2 and see if n is entirely divisible by it. If a remainder is left, we skip the number. Otherwise, we build our factor pair by dividing n by 2 and adding it to our factor pair. We repeat this step for all numbers until we reach the square root we obtained in Step 1.
4. Step 4: At this point, we have the complete factor list. Perform a union on all the numbers in the factor list. The resultant set of numbers are the factors of our original number n.

Example Factorization

Consider the number 20.

1. Step 1: The square root of 20 is 4.47. Rounding it down gives us 4.
2. Step 2: 20 is completely divisible by 1. We add (1, 20) to our factor pair list.
3. Step 3: 20 is completely divisible by 2. We add (2, 10) to our factor pair list.
4. Step 4: 20 is not wholly divisible by 3 as we we get a remainder of 2. So, we skip it.
5. Step 5: 20 is completely divisible by 4. We add (4, 5) to our factor pair list.
6. Step 6: We have reached our square root 4 and don't need to iterate any longer. Our factor pairs are:
   • (1, 20)
   • (2, 10)
   • (4, 5)
7. Step 7: Performing a union on all the numbers in the factor pairs gives us: 1, 2, 4, 5, 10, 20. This list contains all the Factors for the number 20.